For algebraic number fields $K$ with $s\>0$ real and $2t\>0$ complex embeddings and “admissible” subgroups $U$ of the multiplicative group of integer units of $K$ we construct and investigate certain $(s+t)$-dimensional compact complex manifolds $X(K,U)$. We show among other things that such manifolds are non-Kähler but admit locally conformally Kähler metrics when $t=1$. In particular we disprove a conjecture of I. Vaisman.

Contents Introduction 119 1. Quasiregular mappings 120 2. The Beltrami equation 121 3. The Beltrami-Dirac equation 122 4. A quest for compactness 124 5. Sharp $L^p$-estimates versus variational integrals 125 6. Very weak solutions 128 7. Nonlinear commutators 129 8. Jacobians and wedge products 131 9. Degree formulas 134 References 136

We prove a vanishing theorem for the cohomology of the complement of a complex hyperplane arrangement with coefficients in a complex local system. This result is compared with other vanishing theorems, and used to study Milnor fibers of line arrangements, and hypersurface arrangements.

Let M be a real-analytic submanifold of ${ℂ}^n$ whose “microlocal” Levi form has constant rank $s_M^{+}+s_M^{-}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s_M^{-}$, $s_M^{+}$ (and 0).  This phenomenon is known in the literature as “absence of the Poincaré Lemma” and was already proved in case the Levi form is non-degenerate (i.e. $s_M^{-}+s_M^{+}=n-codimM$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively....

We prove that for a real analytic generic submanifold $M$ of ${\mathbb{C}}^n$ whose Levi-form has constant rank, the tangential ${\overline{\partial }}_M$-system is non-solvable in degrees equal to the numbers of positive and $M$ negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with $M$ non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.

Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. For almost complex structures, the existence of a splitting is equivalent to the existence of local coordinates in which the almost complex structure can be represented by a purely numerical matrix, i.e. containing no Grassmann variables. For Riemannian metrics, terms up to degree 2 are allowed in...

Si dimostra con esempi che la distanza di Hausdorff-Carathéodory fra i valori di funzioni multivoche, analitiche secondo Oka, non è subarmonica.

A classical result of Hardy and Littlewood states that if $f\left(z\right)={\sum }_{m=0}^{\infty }{a}_m{z}^m$ is in $H^p$, 0 < p ≤ 2, of the unit disk of ℂ, then ${\sum }_{m=0}^{\infty }{(m+1)}^{p-2}{\left|a_m\right|}^p\le c_p\parallel f{\parallel }_p^p$ where $c_p$ is a positive constant depending only on p. In this paper, we provide an extension of this result to Hardy and weighted Bergman spaces in the unit ball of ${ℂ}^n$, and use this extension to study some related multiplier problems in ${ℂ}^n$.

We introduce the extended bicomplex plane 𝕋̅, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about convergence of sequences of bicomplex meromorphic functions. Hence the concept of normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the fundamental normality tests for families...